Find the Domain (x-8)/(x^2+16x+64)

$\frac{x - 8}{x^{2} + 16 x + 64}$

Step 1

Set the denominator in $\frac{x - 8}{x^{2} + 16 x + 64}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 16 x + 64 = 0$

Step 2

Solve for $x$ .

Factor using the perfect square rule.

Rewrite $64$ as $8^{2}$ .

$x^{2} + 16 x + 8^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 8 + 8^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 8$ .

${(x + 8)}^{2} = 0$

Set the $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 8) \cup (- 8, \infty)$

Set-Builder Notation:

${x | x \neq - 8}$

Step 4